$\dfrac{dy}{dx}=\dfrac{5y}{x}$ Is $y=-2x^5$ a solution to the above equation? Choose 1 answer: Choose 1 answer: (Choice A) A Yes (Choice B) B No
Solution: In order to find whether $y=-2x^5$ is a solution, we need to substitute it into the equation and see if we get equivalent expressions on each side of the equal sign. In addition to substituting for $y$, we need to find the corresponding $\dfrac{dy}{dx}$ expression to substitute into the equation: $\begin{aligned} \dfrac{dy}{dx}&=\dfrac{d}{dx}\left[-2x^5\right] \\\\ &=-10x^4 \end{aligned}$ Now we substitute ${y=-2x^5}$ and ${\dfrac{dy}{dx}=-10x^4}$ into the equation: $\begin{aligned} {\dfrac{dy}{dx}}&=\dfrac{5{y}}{x} \\\\ {-10x^4}&\stackrel{?}{=}\dfrac{5\left({-2x^5}\right)}{x} \\\\ -10x^4&\stackrel{?}{=}\dfrac{-10x^5}{x} \\\\ -10x^4&\stackrel{\checkmark}{=}-10x^4 \end{aligned}$ We obtained the same expression on each side. In conclusion, yes, $y=-2x^5$ is a solution to the differential equation.